Constructing fuzzy controllers with B-spline models - Principles and applications

In this paper we present an approach to designing a novel type of fuzzy controller B spline basis functions are used for input variables and fuzzy singletons for output variables to specify linguistic terms Product is chosen as the fuzzy conjunction and centroid as the defuzzi ca tion method By appropriately designing the rule base a fuzzy controller can be interpreted as a B spline interpolator Such a fuzzy controller may learn to approximate any known data sequences and to minimise a certain cost function By choosing such a function appropriately the learning process can be made to converge rapidly We applied this approach to the problems of function approximation and both supervised and unsupervised learning of mobile robots Experiments validate the advantages of this approach Introduction Recently fuzzy logic control FLC has been successfully applied to a wide range of control problems and has demonstrated some advantages e g in e ciency of developing control software appropriate processing of imprecise sensor data and real time requirements However as pointed out in one obstacle to wide acceptance in industrial applications is that it is still not clear how membership functions defuzzi cation procedures domain discretization and normalization coe cients contribute either in combination or as stand alone factors to the performance of the FLC Two important related issues are Quality of fuzzy controllers In practical applications the smoothness of the controller output is one of the most important design requirements Generally the smoothness can be measured by how many times an output variable can be di erentiated with respect to the input variables The smoothness criterion is applied to the control of very complex systems such as the speed control of automated trains as well as to simple actuators like electrical motors whose life expectancy depends directly on the smoothness of the controller output Unfortunately in general cases a high degree of smoothness cannot be guaranteed and is frequently hard to determine for a given controller Guidelines for choosing membership functions Up to now there exist no convincing guidelines for the successful design of fuzzy controllers This pertains in particular to the choice of a concrete membership function In various fuzzy control applications membership functions of triangular or trapezoidal shape are utilised because of the simplicity of speci cation and the satisfying results But the question still remains Can the control performance be improved by choosing a certain set of membership functions These two issues can be addressed by comparing B spline models with a standard fuzzy logic con troller In our previous work we compared splines and a fuzzy controller with single input single output SISO structures In this paper the multi input single output MISO controller is considered Periodical nonuniform B spline basis functions are interpreted as membership functions MFs Furthermore aspects of function approximation and learning control of mobile robots are discussed Some Previous Work Advances in Fuzzy Control Several authors have shown that fuzzy controllers are universal approximators Wang presents a universal approximator by using Gaussian membership functions product fuzzy conjunction and centroid defuzzi cation Buckley has shown that input output fuzzy controllers are universal approximators Kosko and Dickerson proved that an additive fuzzy system uniformly approximates f X Y if X is compact and f is continuous Two successful applications in commercial controller and process control are given in one is the OMRON temperature controller the other is a gas red water heater The membership functions are selected as triangles and each pair overlaps Can these be generalised as design rules The work in shows that triangular membership functions with a overlap level produce a reconstruction error of zero Further questions are Are there other forms of suitable membership functions Should the overlap of the fuzzy sets for linguistic terms meet certain constraints Popularity of B Splines To solve the problem of numerical approximation for smoothing statistical data basis splines B Splines were introduced by Schoenberg B splines were used later by Riesenfeld and Gordon in Computer Aided Geometric Design CAGD for curve and surface representation Because of their versatility based on only low order polynomials and their straightforward computation B splines have become more and more popular Nowadays B spline techniques represent one of the most important trends in CAD CAM they have been extensively applied in modelling free shape curves and surfaces Recently splines have also been proposed for neural network modelling and control Although fuzzy techniques lend themselves to on line control and B splines have been used mainly in o line modelling some interesting common points can still be found In our previous paper we pointed out that B spline basis functions and parametric membership functions of a linguistic variable are both convex overlapping set functions Splines and fuzzy controllers possess good interpolation features The synthesis of a smooth curve with spline functions can easily be associated with the defuzzi cation process These points are the main motivation for our work on utilising B splines to design fuzzy controllers Construction Principles We consider the membership functions that are used in the context of specifying linguistic terms values or labels of input variables of a fuzzy controller In the following basis functions of periodical Nonuniform B Splines NUBS are summarised and compared with a fuzzy controller We also use B functions for the NUBS basis functions A multi input multi output MIMO rule base is normally divided into several MISO rule bases Synonyms Takagi Sugeno IDM inference and defuzzi cation method Tsukamoto method weighted mean B Spline Basis Functions De ned on a Single Variable Assume x is a general input variable of a control system that is de ned on the universe of discourse x xm Given a sequence of ordered parameters knots x x x xm the ith normalised B spline basis function B function Ni k of order k is de ned as Ni k x for xi x xi otherwise if k x xi xi k xi Ni k x xi k x xi k xi Ni k x if k with i m k The important properties of B functions are Partition of unity Pm i Ni k x Positivity Ni k x Local support Ni k x for x  xi xi k C continuity If the knots fxig are pairwise di erent from each other then Ni k x C i e Ni k x is k times continuously di erentiable Membership Functions of B Function Type The B functions are employed to specify the linguistic terms and knots are chosen to be di erent from each other periodical model Visually the selection of k the order of the B functions determines the following factors of the fuzzy sets for modelling the linguistic terms Table In this table the width of a fuzzy set is measured by the number of knot intervals and the overlap degree by how many fuzzy sets are de ned on each knot interval Order k Degree Shape Rectangular Triangular Quadratic Cubic Fig a Fig b Fig c Fig d Width Overlap Table The visual e ect of fuzzy sets depends mainly on the order of the B functions Real and Virtual Linguistic Terms It is assumed that linguistic terms are to be de ned over x xm the universe of an input variable x of a fuzzy controller They are referred to as real linguistic terms To maintain the partition of unity for all x x xm more B functions should be added at both ends of x xm They are called marginal B functions de ning virtual linguistic terms Real and virtual linguistic terms are illustrated in Figure In the case of order no marginal B function is needed Fig a